Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows

Lai (Geom Topol 25:3629–3690, 2021) used singular Ricci flows, introduced by Kleiner and Lott (Acta Math 219(1):65–134, 2017), to construct a nonnegative Ricci curvature Ricci flow g(t) emerging from an arbitrary 3D complete noncompact Riemannian manifold \((M^3, g_0)\) with nonnegative Ricci curvature. We show g(t) is complete for positive times provided \(g_0\) satisfies a volume ratio lower bound that approaches zero at spatial infinity. Our proof combines a pseudolocality result of Lai (2021) for singular flows, together with a pseudolocality result of Hochard (Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below, 2016. arXiv:1603.08726) and Simon and Topping (J Differ Geom 122(3):467–518, 2022) for nonsingular flows. We also show that the construction of complete nonnegative complex sectional curvature flows by Cabezas-Rivas and Wilking (J Eur Math Soc (JEMS) 17(12):3153–3194, 2015) can be adapted here to show g(t) is complete for positive times provided \(g_0\) is a compactly supported perturbation of a nonnegative sectional curvature metric.